Theorem mobii 2056

Description: Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Hypothesis

Ref	Expression
mobii.1	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
mobii	⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Proof of Theorem mobii

Step	Hyp	Ref	Expression
1		mobii.1	. . . 4 ⊢ (𝜓 ↔ 𝜒)
2	1	a1i 9	. . 3 ⊢ (⊤ → (𝜓 ↔ 𝜒))
3	2	mobidv 2055	. 2 ⊢ (⊤ → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
4	3	mptru 1357	1 ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Syntax hints:   ↔ wb 104  ⊤wtru 1349  ∃*wmo 2020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-eu 2022  df-mo 2023

This theorem is referenced by:  moaneu  2095  moanmo  2096  2moswapdc  2109  2exeu  2111  rmobiia  2659  rmov  2750  euxfr2dc  2915  rmoan  2930  2rmorex  2936  mosn  3619  dffun9  5227  funopab  5233  funco  5238  funcnv2  5258  funcnv  5259  fun2cnv  5262  fncnv  5264  imadif  5278  fnres  5314  ovi3  5989  oprabex3  6108  axaddf  7830  axmulf  7831  frecuzrdgtcl  10368  frecuzrdgfunlem  10375  fsum3  11350